Question

Given the differential equation xy dy/dx = y² + 9, express it in the form ()=(), where ()=?, ()=?, and provide the general solution of the differential equation.

Answer 1

The differential equation xy dy/dx = y² + 9 can be separated to integrate and find the general solution, which is y = 3 tan(3 ln|x| + C).

Step-by-step explanation:

To express the differential equation xy dy/dx = y² + 9 in the form ()=(), where ()=?, ()=?, and to provide the general solution of the differential equation, we must first separate variables and integrate. We can write it as:

dy/y² + 9 = dx/x

Next, we integrate both sides:

∫ dy/(y² + 9) = ∫ dx/x

The left side is a standard integral form that gives:

1/3 arctan(y/3)

The integral on the right side is a natural logarithm:

ln|x| + C

Therefore, equating the two results and solving for the constant C, we get the following general solution:

y = 3 tan(3 ln|x| + C)

This is the solution to the original differential equation.